Black Hole Thermodynamics and Riemann Surfaces
Kirill Krasnov
TL;DR
This work develops a framework to study thermodynamics of 2+1D black holes with negative cosmological constant by analytic continuation to Euclidean handlebodies, linking the partition function to Liouville action and Schottky space moduli. The log partition function acts as a Kaehler potential for the Weil-Petersson metric on the Schottky space, and the on-shell BH entropy is given by a sum over horizon circumferences plus this moduli-dependent term: $S = \sum_i {2\pi r_{+,i}\over 8G} + \ln Z$. For the BTZ case, $\ln Z$ reproduces half the Bekenstein-Hawking entropy, while for general holes the interior geometry nontrivially affects entropy, suggesting a universal bound $I_{\rm Liouv}[\Sigma] \le \sum_i \log|m_i|$. The work connects black hole thermodynamics to rich structures in complex analysis and moduli spaces, with potential implications for bounds and saturation in higher-genus configurations.
Abstract
We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2+1 dimensions. A general black hole in 2+1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g,h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the Kaehler potential for the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black hole entropy leads us to conjecture a new strong bound on this Kaehler potential.